PROGRAMA FONDECYT INFORME FINAL ETAPA 2010 COMISIÓN NACIONAL DE INVESTIGACION CIENTÍFICA Y TECNOLÓGICA VERSION OFICIAL FECHA: 02/11/2010 Nº PROYECTO : 3095018 DURACIÓN : 2 años AÑO ETAPA : 2010 TÍTULO PROYECTO : BLACK HOLES IN HIGHER DIMENSIONS. DISCIPLINA PRINCIPAL : RELATIVIDAD GENERAL Y COSMOLOGIA GRUPO DE ESTUDIO : ASTRON.,COSMOL.Y PAR INVESTIGADOR(A) RESPONSABLE : SOURYA RAY CHAKRABORTY DIRECCIÓN : Edificio Emilio Pugin, Instituto de Ciencias Físicas y Matemáticas COMUNA : CIUDAD : Valdivia REGIÓN : XIV REGION FONDO NACIONAL DE DESARROLLO CIENTIFICO Y TECNOLOGICO (FONDECYT) Moneda 1375, Santiago de Chile - casilla 297-V, Santiago 21 Telefono: 2435 4350 FAX 2365 4435 Email: [email protected] INFORME FINAL PROYECTO FONDECYT POSTDOCTORADO OBJETIVOS Cumplimiento de los Objetivos planteados en la etapa final, o pendientes de cumplir. Recuerde que en esta sección debe referirse a objetivos desarrollados, NO listar actividades desarrolladas. Nº OBJETIVOS CUMPLIMIENTO FUNDAMENTO 1 To study the mechanics and thermodynamics of TOTAL black hole spacetimes. 2 Investigation of symmetries, construction of TOTAL conserved charges and to derive thermodynamical first law type statements using Hamiltonian perturbation techniques. 3 To study the properties of black holes in higher TOTAL dimensions in higher derivative gravity theories. Otro(s) aspecto(s) que Ud. considere importante(s) en la evaluación del cumplimiento de objetivos planteados en la propuesta original o en las modificaciones autorizadas por los Consejos. RESULTS OBTAINED: For each specific goal, describe or summarize the results obtained. Relate each one to work already published and/or manuscripts submitted. In the Annex section include additional information deemed pertinent and relevant to the evaluation process. The maximum length for this section is 5 pages. (Arial or Verdana, font size 10). One of the main objective of the project was to investigate the role of symmetries in black hole spacetimes in higher dimensions. It is well-known that spacetime symmetries manifest themselves in conservation laws, which in turn governs the mechanics of the black holes. Eventhough the mechanics of stationary black holes in Einstein's theory of General Relativity is well understood by now, surprisingly there has not been much progress in understanding the black hole mechanics in presence of a cosmological constant. Moreover, in higher dimensions, there exists a simple generalization of Einstein's General Relativity, known as Lovelock gravity, which allows to explore the effect of higher curvature terms in the black hole thermodynamics. However, in order to fully understand the mechanics of black holes in general Lovelock gravity theories, it is essential to understand the role of the cosmological constant in black hole spacetimes. Another objective of the project was to study black hole spacetimes in higher derivative gravities. Higher derivative terms appear in the low energy effective action of string theory. Generically it is much more difficult to integrate the field equations resulting from these terms. However, one can construct interesting carvature invariants such that for symmetric spacetimes the field equations are integrable. This allows to obtain non-trivial black hole configurations and then study the effect of higher derivatives in the laws of black hole mechanics. These objectives were fulfilled in the following projects. Enthalpy and the Mechanics of AdS Black Holes. One of the objectives of the project was to study the mechanics of black hole spacetimes in presence of a cosmological constant. Black hole solutions with a non-vanishing cosmological constant have received considerable recent attention. This is both due to the role they play in the phenomenology of the AdS/CFT correspondence and to the observational data suggesting that the universe has a small, positive value of Λ. Along with David Kastor and Jennie Traschen, I have studied the mechanics of AdS black holes to bring our understanding more closely in parallel with well known results in the asymptotically flat case. Specifically, we presented a geometric derivation of the Smarr formula for AdS black holes. In asymptotically flat spacetimes, the Smarr formula gives an exact relation between the mass, angular momentum and the entropy of stationary black holes [5]. This formula can be obtained by integrating the Komar integral over all the boundaries of a spatial slice of a stationary black hole spacetime [4]. The Komar integral has recently been generalized for spacetimes with non- vanishing cosmological constant [2]. This is done by adding a Killing potential to the usual Komar integrand. This Killing potential then substracts of the infinite contribution from the stress energy of the cosmological constant within the boundary integral itself, without requiring any regularization procedure via background subtraction. This construction does not rely on the background and gives a sensible answer for relaxed asymptotic falloff conditions. Using this construction we have shown that in asymptotically AdS spacetimes, the Smarr relation gets modified by a new “work” term where the cosmological constant plays the role of pressure and the difference of the boundary integral of the Killing potential evaluated at spatial infinity and at the black hole horizon plays the role of effective volume. Another objective of the project was to investigate the role of symmetries in the construction of conserved charges and to derive first law type statements using Hamiltonian perturbative techniques [3]. We have presented an extended version of the first law of black hole mechanics by including the variations of the cosmological constant. In close analogy to the Smarr relation, the coefficient of the variation of Λ in the first law is determined by the surface integrals of the Killing potential. This coefficient is proportional to a finite, effective volume of the region outside the AdS black hole horizon, which can also be interpreted as minus the volume excluded from a spatial slice by the black hole horizon. Thus, the new term in the first law has the form of effective volume times change in pressure that arises in the variation of the enthalpy in classical thermodynamics. This and related arguments suggest that the mass parameter m of an AdS black hole should be interpreted as the enthalpy of the spacetime. We have further related the expanded first law and the modified Smarr relation by a scaling argument based on Euler's theorem. This work has been published [1]. Smarr Formula and an Extended First Law for Lovelock Gravity. Another objective of the project was to study black holes in alternative theories of gravity. In dimensions greater than four, the simplest generalization of Einstein's theory of General Relativity is Lovelock gravity. In Lovelock gravity, the Lagrangian density is a sum of dimensionally extended Euler densities. Lovelock gravity is the most general second order gravity theory in higher dimensional spacetimes. Along with David Kastor and Jennie Traschen, I have investigated the mechanics of AdS black holes in general Lovelock theories of gravity. In [2], Kastor has generalized the Komar integral for general Lovelock theories. The key ingredient in the construction is a sequence of Killing- Lovelock potentials that are in one-to-one correspondence with the higher curvature terms in the Lagrangian of a Lovelock gravity theory. Using the Hamiltonian pertubation techniques of [3], we have derived an extended first law which includes variations of each of the dimensionful coupling constants bk of Lovelock theory. This introduces new thermodynamic potentials Ψk conjugate to the individual Lovelock couplings bk. We have given a general expression for these potentials. It has been shown that, in general, each of these potentials receives three different contributions. The first is given in terms of the boundary integrals of the Killing-Lovelock potentials. It can be thought of as coming from the effective stress-energy introduced by the higher curvature Lovelock interactions. The second and third contribution come from the explicit dependence of the mass and entropy on the Lovelock couplings. In the simple case of Einstein gravity with a non-zero cosmological constant Λ studied in [1], only first of these contributions arose, because the mass and entropy have no explicit dependence on Λ. Using the extended first law, we have derived the Smarr forumla for static AdS black holes in Lovelock gravity via the scaling argument mentioned previously. The Smarr formula gives the mass of static asymptotically AdS black holes in terms of certain thermodynamic properties, the surface gravity, the entropy, and the potentials Ψk. The Smarr formula becomes particularly useful when such solutions are not known explicitly, as is the case in higher order Lovelock theories. We have also indecated how the Smarr formula may be used in analyzing the Hwaking-Page phase transition in general Lovelock gravity theories. This work has been accepted for publication [6]. A Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions. In this work we continue to investigate exact black hole solutions in higher derivative theories of gravity. Higher derivative theories emerge as correction terms in the effective action in the low energy limit of string theory. However, due to the presence of higher derivatives, they often offer very limited calculational control over the theory. Specifically,